It is common to test many hypotheses simultaneously in the application of statistics. The probability of making a false discovery grows with the number of statistical tests performed. When all the null hypotheses are true, and the test statistics are indepen- dent and continuous, the error rates from the family wise error rate (FWER)- and the false discovery rate (FDR)-controlling procedures are equal to the nominal level. When some of the null hypotheses are not true, both procedures are conservative. In the first part of this study, we review the background of the problem and propose methods to estimate the number of true null hypotheses. The estimates can be used in FWER- and FDR-controlling procedures with a consequent increase in power. We conduct simulation studies and apply the estimation methods to data sets with bio- logical or clinical significance. In the second part of the study, we propose a mixture model approach for the analysis of ChIP-chip high density oligonucleotide array data to study the interac- tions between proteins and DNA. If we could identify the specific locations where proteins interact with DNA, we could increase our understanding of many important cellular events. Most experiments to date are performed in culture on cell lines, bac- teria, or yeast, and future experiments will include those in developing tissues, organs, or cancer biopsies, and they are critical in understanding the function of genes and proteins. Here we investigate the ChIP-chip data structure and use a beta-mixture model to help identify the binding sites. To determine the appropriate number of components in the mixture model, we suggest the Anderson-Darling testing. Our study indicates that it is a reasonable means of choosing the number of components in a beta-mixture model. The mixture model procedure has broad applications in biology and is illustrated with several data sets from bioinformatics experiments.

Test score distributions are used to make important instructional decisions about students. The test scores usually do not follow a normal distribution. In some cases, the scores appear to follow a bimodal distribution that can be modeled with a mixture of beta distributions. This bimodality may be due different levels of students' ability. The purpose of this study was to develop and apply statistical techniques for fitting beta mixtures and detecting bimodality in test score distributions. Maximum likelihood and Bayesian methods were used to estimate the five parameters of the beta mixture distribution for scores in four quizzes in a cell biology class at Brigham Young University. The mixing proportion was examined to draw conclusions about bimodality. We were successful in fitting the beta mixture to the data, but the methods were only partially successful in detecting bimodality.